Integrand size = 37, antiderivative size = 680 \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {8 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {15}{64} b^2 f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {1}{32} b^2 c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}-\frac {4 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {15 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}+\frac {4 i b f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {4 i b c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 f^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {3}{8} f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {1}{4} c^2 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2-\frac {2 i f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {5 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{24 b c \sqrt {1+c^2 x^2}} \]
-8/9*I*b^2*f^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c+15/64*b^2*f^2*x*(d+I* c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)-1/32*b^2*c^2*f^2*x^3*(d+I*c*d*x)^(1/2)*(f-I *c*f*x)^(1/2)-4/27*I*b^2*f^2*(c^2*x^2+1)*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/ 2)/c+3/8*f^2*x*(a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)-1/ 4*c^2*f^2*x^3*(a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)-2/3 *I*f^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2 )/c-15/64*b^2*f^2*arcsinh(c*x)*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c/(c^2* x^2+1)^(1/2)+4/3*I*b*f^2*x*(a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x )^(1/2)/(c^2*x^2+1)^(1/2)-3/8*b*c*f^2*x^2*(a+b*arcsinh(c*x))*(d+I*c*d*x)^( 1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+4/9*I*b*c^2*f^2*x^3*(a+b*arcsinh( c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/8*b*c^3*f^2* x^4*(a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/ 2)+5/24*f^2*(a+b*arcsinh(c*x))^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/b/c/( c^2*x^2+1)^(1/2)
Time = 3.73 (sec) , antiderivative size = 890, normalized size of antiderivative = 1.31 \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {6912 i a b c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}-4608 i a^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-6912 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2592 a^2 c f^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4608 i a^2 c^2 f^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-1728 a^2 c^3 f^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1440 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-1728 a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-256 i b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+108 a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+4320 a^2 \sqrt {d} f^{5/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+864 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+768 i a b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))-27 b^2 f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (4 \text {arcsinh}(c x))+12 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (576 i b c x-576 i a \sqrt {1+c^2 x^2}-144 b \cosh (2 \text {arcsinh}(c x))-192 i a \cosh (3 \text {arcsinh}(c x))+9 b \cosh (4 \text {arcsinh}(c x))+288 a \sinh (2 \text {arcsinh}(c x))+64 i b \sinh (3 \text {arcsinh}(c x))-36 a \sinh (4 \text {arcsinh}(c x))\right )+72 b f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (60 a-48 i b \sqrt {1+c^2 x^2}-16 i b \cosh (3 \text {arcsinh}(c x))+24 b \sinh (2 \text {arcsinh}(c x))-3 b \sinh (4 \text {arcsinh}(c x))\right )}{6912 c \sqrt {1+c^2 x^2}} \]
((6912*I)*a*b*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (4608*I)*a^2*f ^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (6912*I)*b^2*f^ 2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2592*a^2*c*f^2*x *Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (4608*I)*a^2*c^2* f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 1728*a^2*c ^3*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1440*b^ 2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 1728*a*b*f^2*Sq rt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - (256*I)*b^2*f^2*S qrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 108*a*b*f^2*Sqrt [d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 4320*a^2*Sqrt[d]*f^ (5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sq rt[f - I*c*f*x]] + 864*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2* ArcSinh[c*x]] + (768*I)*a*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3 *ArcSinh[c*x]] - 27*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*Arc Sinh[c*x]] + 12*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((5 76*I)*b*c*x - (576*I)*a*Sqrt[1 + c^2*x^2] - 144*b*Cosh[2*ArcSinh[c*x]] - ( 192*I)*a*Cosh[3*ArcSinh[c*x]] + 9*b*Cosh[4*ArcSinh[c*x]] + 288*a*Sinh[2*Ar cSinh[c*x]] + (64*I)*b*Sinh[3*ArcSinh[c*x]] - 36*a*Sinh[4*ArcSinh[c*x]]) + 72*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a - (48*I )*b*Sqrt[1 + c^2*x^2] - (16*I)*b*Cosh[3*ArcSinh[c*x]] + 24*b*Sinh[2*Arc...
Time = 1.24 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.50, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6211, 27, 6253, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \int f^2 (1-i c x)^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \int (1-i c x)^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \int \left (-c^2 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-2 i c x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {1}{8} b c^3 x^4 (a+b \text {arcsinh}(c x))+\frac {4}{9} i b c^2 x^3 (a+b \text {arcsinh}(c x))+\frac {3}{8} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {2 i \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {1}{4} c^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {3}{8} b c x^2 (a+b \text {arcsinh}(c x))+\frac {4}{3} i b x (a+b \text {arcsinh}(c x))+\frac {5 (a+b \text {arcsinh}(c x))^3}{24 b c}-\frac {15 b^2 \text {arcsinh}(c x)}{64 c}+\frac {15}{64} b^2 x \sqrt {c^2 x^2+1}-\frac {4 i b^2 \left (c^2 x^2+1\right )^{3/2}}{27 c}-\frac {8 i b^2 \sqrt {c^2 x^2+1}}{9 c}-\frac {1}{32} b^2 c^2 x^3 \sqrt {c^2 x^2+1}\right )}{\sqrt {c^2 x^2+1}}\) |
(f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((((-8*I)/9)*b^2*Sqrt[1 + c^2*x^2 ])/c + (15*b^2*x*Sqrt[1 + c^2*x^2])/64 - (b^2*c^2*x^3*Sqrt[1 + c^2*x^2])/3 2 - (((4*I)/27)*b^2*(1 + c^2*x^2)^(3/2))/c - (15*b^2*ArcSinh[c*x])/(64*c) + ((4*I)/3)*b*x*(a + b*ArcSinh[c*x]) - (3*b*c*x^2*(a + b*ArcSinh[c*x]))/8 + ((4*I)/9)*b*c^2*x^3*(a + b*ArcSinh[c*x]) + (b*c^3*x^4*(a + b*ArcSinh[c*x ]))/8 + (3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/8 - (c^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/4 - (((2*I)/3)*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/c + (5*(a + b*ArcSinh[c*x])^3)/(24*b*c)))/Sqrt[1 + c^ 2*x^2]
3.6.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ ) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 + c^2*x ^2)^q) Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^ 2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
\[\int \left (-i c f x +f \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {i c d x +d}d x\]
\[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {i \, c d x + d} {\left (-i \, c f x + f\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
integral(-(b^2*c^2*f^2*x^2 + 2*I*b^2*c*f^2*x - b^2*f^2)*sqrt(I*c*d*x + d)* sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1))^2 - 2*(a*b*c^2*f^2*x^2 + 2 *I*a*b*c*f^2*x - a*b*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + s qrt(c^2*x^2 + 1)) - (a^2*c^2*f^2*x^2 + 2*I*a^2*c*f^2*x - a^2*f^2)*sqrt(I*c *d*x + d)*sqrt(-I*c*f*x + f), x)
Timed out. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Timed out} \]
Exception generated. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {d+i c d x} (f-i c f x)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{5/2} \,d x \]